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An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

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Abstract

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v t = v x v y - ∂ x -1 ∂ y [v y + v x 2], where the formal integral ∂ x −1 becomes the asymmetric integral \( - \int_x^\infty {dx'} \) . We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f(X, Y) over a parabola in the plane (X, Y) can be expressed in terms of the integrals of f(X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Chernogolovka, Russia

    P. G. Grinevich

  2. Lomonosov Moscow State University, Moscow, Russia

    P. G. Grinevich

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia

    P. G. Grinevich

  4. Dipartimento di Fisica, Università di Roma “La Sapienza,”, Rome, Italy

    P. M. Santini

  5. Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome, Italy

    P. M. Santini

Authors
  1. P. G. Grinevich
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  2. P. M. Santini
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Corresponding author

Correspondence to P. G. Grinevich.

Additional information

The research of P. G. Grinevich was supported in part by the Russian Foundation for Basic Research (Grant No. 13-01-12469 ofi m2), the Program for Supporting Leading Scientific Schools (Grant No. NSh-4833.2014.1), the program “Fundamental problems of nonlinear dynamics” of the Presidium of the Russian Academy of Sciences, the INFN sezione di Roma, and the program PRIN 2010/11 (Program No. JJ4KPA 004 of Roma 3).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 1, pp. 59–68, October, 2016.

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Grinevich, P.G., Santini, P.M. An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects. Theor Math Phys 189, 1450–1458 (2016). https://doi.org/10.1134/S0040577916100056

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  • DOI: https://doi.org/10.1134/S0040577916100056

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